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(A) We want to find the minimum value of $f(	heta) = 3 \sin 	heta + 	ext{cosec}^3 	heta$ for $0 < 	heta < \pi$.Using the $A.M. \ge G.M.$ inequali

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(A) We want to find the minimum value of $f(	heta) = 3 \sin 	heta + 	ext{cosec}^3 	heta$ for $0 Using the $A.M. \ge G.M.$ inequality for four positive terms: $\sin 	heta, \sin 	heta, \sin 	heta, 	ext{cosec}^3 	heta$.$\frac{\sin 	heta + \sin 	heta + \sin 	heta + 	ext{cosec}^3 	heta}{4} \ge \sqrt[4]{(\sin 	heta \cdot \sin 	heta \cdot \sin 	heta \cdot 	ext{cosec}^3 	heta)}$$\frac{3 \sin 	heta + 	ext{cosec}^3 	heta}{4} \ge \sqrt[4]{(\sin^3 	heta \cdot 	ext{cosec}^3 	heta)}$Since $\sin 	heta \cdot 	ext{cosec} 	heta = 1$,we have:$\frac{3 \sin 	heta + 	ext{cosec}^3 	heta}{4} \ge \sqrt[4]{1^3} = 1$$3 \sin 	heta + 	ext{cosec}^3 	heta \ge 4$Thus,the minimum value is $4$.

If $0 < \theta < \pi$,then the minimum value of $3 \sin \theta + \text{cosec}^3 \theta$ is

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